We study first-order logic over unordered structures whose elements carry a finite number of data values from an infinite domain. Data values can be compared wrt.\ equality. As the satisfiability problem for this logic is undecidable in general, we introduce a family of local fragments. They restrict quantification to the neighbourhood of a given reference point that is bounded by some radius. Our first main result establishes decidability of the satisfiability problem for the local radius-1 fragment in presence of one ``diagonal relation''. On the other hand, extending the radius leads to undecidability. In a second part, we provide the precise decidability and complexity landscape of the satisfiability problem for the existential fragments of local logic, which are parameterized by the number of data values carried by each element and the radius of the considered neighbourhoods. Altogether, we draw a landscape of formalisms that are suitable for the specification of systems with data and open up new avenues for future research.

翻译：我们研究无序结构上的一阶逻辑，其中元素携带来自无限域的有限个数据值。数据值之间可进行相等性比较。由于该逻辑的可满足性问题在一般情况下不可判定，我们引入了一族局部片段。这些片段将量词限制在以给定参考点为中心、半径有界的邻域内。我们的第一个主要结果建立了含单条“对角关系”的局部半径-1片段可满足性问题的可判定性。另一方面，扩展半径将导致不可判定性。在第二部分中，我们给出了局部逻辑存在片段可满足性问题的精确可判定性与复杂性全景图——该问题以每个元素携带的数据值数量和所考虑邻域半径为参数。综上，我们描绘了一组适用于含数据系统规约的形式化方法，并为未来研究开辟了新的方向。